A moment map for the action of a super Lie group \(G\)  on a symplectic supermanifold \(M\)  is a map \(\mu \colon M \rightarrow \mathfrak {g}^* = \mathrm {Lie}(G)^*\)  of Poisson supermanifolds such that the infinitesimal action of an element \(a\in \mathfrak {g}\)  is given by the Hamiltonian vector field for the function \(m \mapsto \langle \mu (m),a\rangle \) , where \(\langle \,,\,\rangle \)  denotes the pairing of the vector space \(\mathfrak {g}\)  with its dual; such data is equivalent to a Lie algebra map \(\mu ^*\colon \mathfrak {g}\rightarrow \mathcal {O}(M)\) , where the latter is treated as a super Lie algebra with its Poisson bracket. When certain cohomology groups of \(M\)  and \(\mathfrak {g}\)  vanish, \(\mu \)  exists and is unique. The Hamiltonian reduction \(M // G\)  of this data is the quotient space \(\mu ^{-1}(0) / G\) . This can be defined in the super case via its algebra of functions \((\mathcal {O}(M)/\langle \mu ^*\mathfrak {g}\rangle )^G\) , where \(\langle \mu ^*\mathfrak {g}\rangle \)  denotes the ideal generated by the image of \(\mu ^*\) . As Marsden and Weinstein explained in the even case [1]}, when 0 is a regular value of \(\mu \)  and the action of \(G\)  on \(\mu ^{-1}(0)\)  is free and proper, the manifold \(M // G\)  is naturally symplectic. The natural maps \(\mu ^{-1}(0) M\)  and \(\mu ^{-1}(0) M//G\)  are together a Lagrangian correspondence between \(M\)  and \(M//G\) . Super Hamiltonian reduction can be cleanly expressed as an example of coisotropic reduction of super Poisson algebras [2]}, [3]}.
